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The Journal of Neuroscience, April 15, 1999, 19(8):3122-3145
A Theory of Geometric Constraints on Neural Activity
for Natural Three-Dimensional Movement
Kechen
Zhang1 and
Terrence J.
Sejnowski1, 2
1 Howard Hughes Medical Institute, Computational
Neurobiology Laboratory, The Salk Institute for Biological Studies,
La Jolla, California 92037, and 2 Department of Biology,
University of California, San Diego, La Jolla, California 92093
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ABSTRACT |
Although the orientation of an arm in space or the static view of
an object may be represented by a population of neurons in complex
ways, how these variables change with movement often follows simple
linear rules, reflecting the underlying geometric constraints in the
physical world. A theoretical analysis is presented for how such
constraints affect the average firing rates of sensory and motor
neurons during natural movements with low degrees of freedom, such as a
limb movement and rigid object motion. When applied to nonrigid
reaching arm movements, the linear theory accounts for cosine
directional tuning with linear speed modulation, predicts a curl-free
spatial distribution of preferred directions, and also explains why the
instantaneous motion of the hand can be recovered from the neural
population activity. For three-dimensional motion of a rigid object,
the theory predicts that, to a first approximation, the response of a
sensory neuron should have a preferred translational direction and a
preferred rotation axis in space, both with cosine tuning functions
modulated multiplicatively by speed and angular speed, respectively.
Some known tuning properties of motion-sensitive neurons follow as
special cases. Acceleration tuning and nonlinear speed modulation are
considered in an extension of the linear theory. This general approach
provides a principled method to derive mechanism-insensitive neuronal
properties by exploiting the inherently low dimensionality of natural movements.
Key words:
3-D object; cortical representation; visual cortex; tuning curve; motor system; reaching movement; speed modulation; potential function; gradient field; zero curl
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INTRODUCTION |
For natural movements, such as the
motion of a rigid object or an active limb movement, many sensory
receptors or muscles are involved, but the actual degrees of freedom
are low because of geometric constraints in the physical world. For
example, as illustrated in Figure 1, the
rotation of an object alters many visual cues. How these cues vary in
time is not arbitrary but is fully determined by the rigid motion,
which has only 6 degrees of freedom. As a consequence, neuronal
activity reflecting such natural movements also is likely to be highly
constrained and to have only a few degrees of freedom.
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Figure 1.
Axis of rotation determines how the view of an
object changes instantaneously, along with various visual cues, such as
shading, shadow, mirror reflection, glare, and occlusion. Rigid
geometry predicts that the response of a motion-sensitive neuron, to a
first approximation, should have a preferred rotation axis in
three-dimensional space with cosine tuning function and linear angular
speed modulation, regardless of the exact cues used and the exact
computational mechanisms involved.
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This paper presents a theoretical analysis of how neuronal activity
correlated with natural movements might be constrained by geometry. The
basic theory, although essentially linear, can account for several key
features of diverse neurophysiological results and generates strong
predictions that are testable with current experimental techniques.
An emerging principle from this analysis is that neuronal activity
tuned to movement often obeys simple generic rules as a first
approximation, insensitive to the exact sensory or motor variables that
are encoded and the exact computational interpretation. Such generic
tuning properties are mechanism insensitive because they are
better described as reflecting the underlying geometric constraints on
movements rather than the actual computational mechanisms. This
simplicity arises when sensory or motor variables represent changes in
time rather than static values. In the example shown in Figure 1, the
viewpoint was fixed and the object was rotated systematically
around different axes. The focus is on how neuronal responses depend on
the rotation axis in three-dimensional space, given approximately the
same view of the object. It is possible to derive a simple cosine
tuning rule for the rotation axis, although various visual cues may
depend on the static geometrical orientation of the object in complex
ways. Three-dimensional object motion is a specific example; the same
principles also apply to several other biological systems, including
nonrigid arm movement.
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DIRECTIONAL TUNING FOR ARM MOVEMENT |
Although the visual and the motor examples share similar
mechanism-insensitive properties, the reaching arm movement has a simpler mathematical description and more supporting experimental results and will be considered first.
Ubiquity of cosine tuning
A directional tuning curve describes how the mean firing rate of a
neuron depends on the reaching direction of the hand. As illustrated in
Figure 2, broad cosine-like tuning curves
are very typical in many areas of the motor system of monkeys,
including the primary motor cortex (Georgopoulos et al., 1986 ),
premotor cortex (Caminiti et al., 1991), parietal cortex (Kalaska et
al., 1990), cerebellum (Fortier et al., 1989), basal ganglia (Turner and Anderson, 1997), and somatosensory cortex (Cohen et al., 1994; Prud'homme and Kalaska, 1994). Although the examples shown in Figure 2
are two-dimensional, cosine tuning holds as well for three-dimensional
reaching movement (Georgopoulos et al., 1986; Schwartz et al.,
1988).
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Figure 2.
Cosine tuning to hand movement direction is very
common in monkey motor system, here showing examples of average tuning
curves in two-dimensional reaching tasks, with preferred direction
taken as 0°. Left column, Circular normal functions
(solid curves) fit the data ( ) slightly better and are
slightly narrower than cosine functions (dashed curves).
Horizontal lines indicate background firing rates without
movement. Right column, Data and the circular normal
functions after subtracting the cosine functions. Data from motor
cortex (M1) and cerebellum (Purkinje cells plus deep nuclei) are from
Figure 2 in Fortier et al. (1993), basal ganglia data (GPe) are from
Figure 8B (decrease type) in Turner and Anderson (1997), and
somatosensory cortex data (S1) are from Figure 11A (no load case) in
Prud'homme and Kalaska (1994), with permission.
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The ubiquity of cosine tuning is a hint that this property is generic
and insensitive to the exact computational function of these neurons.
For example, coding of muscle shortening rate is one theoretical
mechanism that can generate cosine tuning (Mussa-Ivaldi, 1988). As
another example, many somatosensory cortical cells related to reaching
had cosine directional tuning, probably because of the geometry of
mechanical deformation of the skin during arm movement (Cohen
et al., 1994; Prud'homme and Kalaska, 1994). Because a cosine tuning
function implies a dot product between a fixed preferred direction and
the actual reaching direction (Georgopoulos et al., 1986), cosine
tuning by itself suggests a linear relation with reaching direction
(Sanger, 1994), which could arise as an approximation to the activity
in a nonlinear recurrent network (Moody and Zipser, 1998). Therefore,
cosine tuning curves should be common in a theoretical model that is
approximately linear.
Basic theory
In this section we derive a general tuning rule for motor neurons
and then discuss its basic properties. This example illustrates what is
meant by mechanism-insensitive properties and the general theoretical
argument based on geometric constraints.
Consider stereotyped reaching movement in which the configuration of
the whole arm is determined completely by the hand position (x,
y, z) in space. In other words, such movements have only 3 degrees
of freedom. Assume that the mean firing rate of a neuron relative
to baseline is proportional to the time derivative of an unknown smooth
function of hand position in space. In other words:
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(1)
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where f is the firing rate, f0
is the baseline rate, and  is an arbitrary function of the hand
position (x, y, z). A possible small time difference between
the neural activity and the arm movement may also be included, as appropriate.
The function (x, y, z) could have any form and could
include any function of arm configuration, such as muscle length, joint angles, or any combination of those. Mussa-Ivaldi (1988) first used
muscle length to demonstrate the appearance of cosine tuning in a
two-dimensional situation and pointed out that the argument could be
generalized to include other muscle variables. This interesting example
illustrates how cosine tuning property might emerge from some simple
assumptions. The assumption in Equation 1 is more general and the
formalism is simpler than that of Mussa-Ivaldi (1988) because joint
angles are no longer used as intermediate variables in the derivation.
This makes interpretation easier and more flexible and the curl-free
condition more apparent (see below). The precise interpretation of is not the focus of this paper; the only requirement is that it be a
function fully determined by the hand position in the three-dimensional space.
We emphasize that although Equation 1 uses hand position as the only
free variables, this does not require that the neuron must
directly encode the hand position or end-point in particular or
kinematic variables in general. Stereotypical reaching movements have
only 3 degrees of freedom and can be conveniently parameterized by the
hand position (x, y, z), although other parameters can also
be used without affecting the final conclusion (see below and Appendix
A). A neuron related to reaching arm movement should be sensitive to
changes of arm posture, which can always be expressed equivalently as
changes in some functions of the hand position (x, y, z).
The simplest estimate of such changes is the first temporal derivative
given in Equation 1. In other words, the above assumption only
postulates a general dependence of the firing rate of a neuron on
changing arm posture as a first approximation, regardless of which
parameters are encoded and how they are encoded.
The assumption in Equation 1 implies that the mean firing rate of a
neuron should follow the tuning rule:
|
(2)
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where v = ( ,  ,  ) is the
instantaneous reaching velocity of the hand, and the vector
p is the preferred reaching direction, given by:
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(3)
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The derivation of this result follows immediately from the chain
rule:
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(4)
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For hand movements starting from the same position (x, y,
z) in space, the tuning rule in Equation 2 implies cosine
directional tuning and linear speed modulation (see Eq. 12). The
preferred direction vector p = p(x, y,
z) of the neuron may depend on the starting hand position. It can
be regarded as a constant vector when the hand is close to its starting position.
For hand movements starting from different positions, the preferred
direction vector may vary with the starting hand position (x, y,
z) and thus can be visualized as a vector field (Caminiti et al.,
1990; Moody and Zipser, 1998). It follows from the gradient formula in
Equation 3 that this vector field of preferred direction must have zero
curl:
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(5)
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because of the equality of mixed second partial derivatives of
. This means that the components of the preferred direction cannot
vary arbitrarily with the starting hand position. An equivalent integral formulation of the curl-free condition is that the path integral of p vanishes along any closed curve in
three-dimensional space:
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(6)
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with dl = (dx, dy, dz), assuming that there
are no singularities in the vector field. This constrains how the
preferred direction of a neuron should vary with the starting hand
position. Any distribution with non-zero curl can be ruled out (Fig.
3).
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Figure 3.
The preferred direction field of a hypothetical
neuron that violates the curl-free condition in a planar reaching task.
For each hand position, the preferred direction of this neuron is
always perpendicular to the straight line from the hand
(H) to the shoulder (S), and the length
of the vector is proportional to the distance of HS. This
vector field has constant non-zero curl everywhere in the work space.
The gradient theory does not allow the existence of such a
neuron.
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Human eyes are not reliable at judging whether a vector field is
curl-free (see Fig. 10), so numerical computation is needed (Mussa-Ivaldi et al., 1985; Giszter et al., 1993). See Appendix B for
more discussion. A vector field is curl-free if and only if it can be
generated as the gradient of a potential function. A more intuitive
interpretation of the curl-free condition is that when a vector field
is regarded as the velocity field of a fluid, there is no net
circulation along any closed path in space.
Under the curl-free condition, the net spike count (integration of the
firing rate with respect to baseline over time) can be used to recover
the value of the unknown potential function:
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(7)
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where the integral depends only on the initial hand position
(x0, y0,
z0) at time 0 and the final position (x,
y, z) at time T, not on the exact trajectory of hand
movement. For each hand position, the firing rate is the largest when
the hand moves along the local gradient of the potential function,
which defines p.
Baseline firing rate
The theory in the preceding section does not constrain the
baseline firing rate f0, which needs to
be considered separately. By definition, the baseline firing rate is
independent of the reaching direction, but it may be modulated by
several other factors. For example, in the motor cortex, Kettner et al.
(1988) have reported that the linear formula:
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(8)
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approximately described the baseline firing rate while the hand
was held fixed at position (x, y, z) in the
three-dimensional work space, where a0,
a1, a2, a3 are
constant coefficients. For reaching at speed v, a more
general linear formula for the baseline firing rate is:
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(9)
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where the coefficients a0,
a1, a2, a3,
a are independent of the hand position (x, y, z) and
the speed v, but may vary with task conditions. For
instance, the baseline firing rate when the hand is held still (Fig. 2,
horizontal lines) differs from the baseline rate defined as
the average of the cosine curve during reaching. Moran and Schwartz
(1999) showed that a linear speed term for baseline rate should
be included in the fitting formula, although their analysis used the
square root of firing rate instead of the raw firing rate. Indirect
evidence for a linear speed term in baseline rate is provided by the
linear effect of reaching distance (see below).
Note that in Equation 9, the baseline firing rate contains information
about both the static hand position (x, y, z) and its speed
v. As shown by Kettner et al. (1988), the spatial gradient of the spontaneous firing rate for static hand position tends to be
consistent with the preferred direction of the same neuron. In the
current theory, this means that the preferred direction p =  tends to point in the same direction as the
vector (a1, a2,
a3) in Equation 9. Therefore, if the potential
function = (x, y, z) can be approximated as a
linear function in x, y, z, we can replace Equation 9
by:
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(10)
|
where k is a constant coefficient. In this case, the
overall firing rate of a neuron that obeys the basic tuning rule in Equation 2 would convey two pieces of information: the baseline firing
rate f0 would represent the static value of the
potential function , and the directionally tuned part
p · v would represent the spatial gradient of
the same potential function.
Linear theory without gradient
If we simply postulate that the firing rate of a neuron is
linearly related to the components of reaching velocity v = (vx, vy,
vz), we would have the same tuning rule:
|
(11)
|
where the components of the preferred direction,
(px, py,
pz)  p, are three arbitrary functions
of the hand position (x, y, z). For a single starting hand
position, this tuning rule is locally indistinguishable from the
prediction of the gradient theory. The difference is that now the
preferred direction field is not required to be the gradient of any
potential function so that its global distribution in hand position
space is not constrained at all. In other words, this vector field need
not be curl-free. The nongradient theory is more general, allowing a
circular distribution of the preferred directions as in Figure 3. The
necessary and sufficient condition for the gradient theory to be true
is that the preferred direction field is curl-free. The existing data cannot distinguish the two theories (see discussion below and Appendix
B).
Comparison with experimental results
Data from a wide range of motor-related brain areas largely
confirm the tuning rule in Equation 2 as a reasonable approximation, together with its various ramifications as follows. Theoretical predictions such as the curl-free distribution remain to be tested.
Cosine directional tuning and multiplicatively linear
speed modulation
The tuning rule in Equation 2 captures two main effects: cosine
directional tuning and multiplicatively linear speed modulation, as
clearly seen in its equivalent form:
|
(12)
|
where v = |v| is the reaching speed, the
proportional factor p = |p| is length of the
preferred direction vector p, and  is the angle between
the instantaneous reaching direction and the preferred direction.
Because the hand trajectory is approximately straight in normal
reaching, the instantaneous velocity v is a vector that
points in the same direction as the reaching direction. If a tuning
function is cosine in three-dimensional space, it must also be cosine
in any two-dimensional subspace, as in the examples in Figure 2.
A cosine function is a good approximation to the directional tuning
data, although a circular normal function (Eq. A17), with one more free
parameter, tends to fit the data slightly better (Fig. 2). The residual
can be roughly accounted for by an additional Fourier term, cos 2,
with an amplitude less than ~10% of that of the original term, cos
(see further discussion in Appendix A).
The speed modulation effect predicted by Equation 12 is multiplicative;
that is, the firing rate should be higher for faster reaching speed
without affecting the shape of the cosine tuning function. This is
approximately true as shown by Moran and Schwartz (1999), who,
however, used the square root of firing rate in analysis so that the
linearity of speed modulation on raw firing rate was not directly
quantified. Indirect evidence for linear speed modulation includes
trajectory reconstruction and the curvature power law (see below).
Neuronal population vector
Suppose the firing rate of each neuron i in a
population (i = 1, 2, ... , N) follows the same
tuning rule as considered above:
|
(13)
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The population vector u is defined as the vector sum of
the preferred directions pi weighted by
firing rates relative to baselines (Georgopoulos et al., 1986):
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(14)
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where in the second step, Equation 13 is used. For the population
vector u to be proportional to the true velocity v, namely:
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(15)
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the necessary and sufficient condition is that the preferred
directions satisfy:
|
(16)
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where  is an arbitrary constant, I is 3 × 3 identity matrix, each pi is a column
vector, and piT is a row vector
(Mussa-Ivaldi, 1988; Gaál, 1993; Salinas and Abbott, 1994;
Sanger, 1994). In particular, when pi are
distributed uniformly, as is roughly true for cells in motor cortex
(Georgopoulos et al., 1988), the condition in Equation 16 is satisfied
so that Equation 15 follows as a consequence. Then the population
vector approximates the reaching direction and reaching velocity (Moran
and Schwartz, 1999).
Trajectory reconstruction
One implication of Equation 15 is that integration over the
population over time can reconstruct the hand trajectory, up to a
scaling constant:
|
(17)
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where r(t) is hand position at time t.
This is consistent with the finding that adding up the population
vector head-to-tail approximately reproduced the shape of the hand
trajectory (Schwartz, 1993, 1994), because head-to-tail addition is a
discrete approximation to the continuous vector integration.
Curvature power law
While drawing, the hand moves more slowly when the trajectory is
more highly curved, and obeys a power law:
|
(18)
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where  is instantaneous angular velocity with respect to an
instantaneous center determined by the local curvature  of the
trajectory, and B is a constant (Lacquaniti et al., 1983). Schwartz (1994) showed that the changing direction of the population vector of cells in motor cortex of monkeys followed the same power law
during drawing. This is consistent with Equation 15, which requires
that the population vector u be proportional to the
instantaneous hand velocity v, up to a possible time difference. This form of the power law involves only the direction of
population vector u. To test its length u = |u| or the linearity of firing rate modulation by
reaching speed, one may use the equivalent form of the power law:
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(19)
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where v = r is the hand speed and r = 1/ is the local radius for the curvature of the trajectory.
The length of the population vector u is proportional to the
hand speed v if and only if the population vector follows
the same power law in Equation 19.
Reaching distance
Fu et al. (1993) reported a nearly linear correlation between
firing rates of cells in motor cortex and reaching distance. Although
this result was somewhat confounded by faster reaching for longer
distances, it raises the question of the general effect of reaching
distance. A linear distance effect would be consistent with the basic
model in Equation 2, which implies that:
|
(20)
|
where vector d = r(T)  r(0) is the final
displacement from the starting position r(0), assuming the
preferred direction p is approximately constant along movement trajectory. The dot product p · d
implies a linear relation between the reaching distance and the total spike count above baseline, together with a cosine directional tuning,
regardless of the exact time course of hand velocity.
Note that in Equation 20 the baseline rate f0
has been subtracted. Because the baseline rate itself may contain a
linear speed component as in Equation 9 (Moran and Schwartz, 1999), its
contribution to total spike count should be:
|
(21)
|
where, for simplicity, a1 = a2 = a3 = 0 has been assumed to ignore the effect of static
hand position. Because the last term is proportional to the reaching
distance |d| but independent of the reaching direction,
it might account for the observation that the modulation of overall
firing rates by reaching distance was often linear but insensitive to
the reaching direction (Fu et al., 1993; Turner and Anderson,
1997).
Curl-free distribution of preferred direction
Caminiti et al. (1990, 1991) reported that the preferred direction
of a motor cortical neuron often varied with the starting point of hand
movement. This is allowed by the gradient theory, provided that this
vector field is curl-free, according to Equation 5 or 6. A constant
preferred direction field is always allowed because it has zero curl.
The curl-free condition constrains how the preferred direction of a
neuron may vary in different parts of space. For example, it rules
out the possibility of any circular arrangement of the preferred
directions, such as that in the two-joint planar arm example shown in
Figure 3. The existing data do not include enough points to compute the
curl (see Appendix B). Further experiments would be needed to test
whether the prediction of the gradient theory is correct.
Elbow position
Scott and Kalaska (1997) found that the preferred directions of
some motor cortical cells were altered when the monkey had to reach
unnaturally with the elbow raised to shoulder level. In the current
theoretical framework, adding elbow position as a free parameter is
equivalent to adding one rotation variable  , for example, the angle
between the horizontal plane and the plane determined by the hand,
elbow, and shoulder. The same theoretical argument yields the tuning
rule:
|
(22)
|
where K is a coefficient that may depend on both hand
position and elbow position. This formula implies two new effects. The
first is that now the preferred direction vector, both its direction
and length, may depend on the elbow position as well as the hand
position (x, y, z):
|
(23)
|
as reported by Scott and Kalaska (1997). The second effect, a new
prediction, is that the firing rate may contain a component proportional to the angular speed d/dt of elbow rotation.
How does this case relate to our earlier results with hand position as
the only free parameter? In the preceding sections, reaching was
assumed to be "stereotypical" in the sense that the elbow position
can be determined completely by the hand position, ignoring forearm
rotation. This assumption may not be true if the final posture
sometimes depends also on the initial hand position (Soechting et al.,
1995). However, when comparing reaching movements starting from the
same initial hand position, it is reasonable to assume that for
stereotypical reaching, the elbow angle can be completely
determined by the hand position (x, y, z), or = (x, y, z). Then the time derivative of , after expanding by the chain rule, can be absorbed into the term
p · v, yielding the original basic tuning rule
in Equation 2. In other words, the assumption of stereotypical movement
reduces the total degrees of freedom to 3, eliminating the elbow
position as an independent variable. Although the elbow angle can still
be used as a free parameter, it is no longer independent of the hand
position. Only three parameters are independent in this case, and their exact choice does not affect the general form of the tuning rule (see
Appendix A for more discussion on coordinate-system independence).
Summary and discussion of more complex cases
As shown above, the basic tuning theory can naturally account for
several important experimental results without making any specific
assumptions about the exact variables encoded or details of the
encoding. These results are generic properties independent of the exact
functional interpretations. This generality makes sense because during
stereotypical movement, redundant variables are inevitably
constrained by the geometry and become highly correlated, so that
they are likely to show similar tuning properties of the same
general type. The theory presented here has formalized this intuition.
The relationship between cosine tuning properties and geometric
constraints is also apparent in the studies of muscle activities and
actions during reaching and isometric tasks. Basic properties resembling those for motor cortical cells have been reported, including
approximately cosine directional tuning curves (but often with a small
secondary peak opposite the preferred direction), speed sensitivity,
and posture dependence (Flanders and Soechting, 1990; Flanders and
Herrmann, 1992; Buneo et al., 1997).
The basic theory needs to be generalized in situations where the hand
position is not the only free parameter. For example, force is one
variable that is often correlated with the activity of motor cortex;
recent examples related to directional tuning include tasks with static
load (Kalaska et al., 1989) and varying isometric forces (Georgopoulos
et al., 1992; Sergio and Kalaska, 1997).
As another example, preparatory activity in motor cortex before onset
of movement can reflect the upcoming reaching direction, as is
especially evident during instructed delay (Georgopoulos et al.,
1989a), and can change rapidly in tasks requiring mental rotation
(Georgopoulos et al., 1989b) or target switching (Pellizzer et al.,
1995).
Moreover, when sensory and motor components were decoupled, some
neurons even from primary motor cortex were more closely related to the
visual movement of a cursor on the computer screen than to the joystick
position or hand movements, in both one-dimensional (Alexander and
Crutcher, 1990) and two-dimensional tasks (Shen and Alexander, 1997a).
By contrast, in virtual reality experiments with visual distortion,
motor cortical activity mainly followed the actual limb trajectory
rather than the animal's visual perception (Moran et al., 1995).
In addition, some differences exist among the neural activity from
different brain areas, although they all show approximate cosine
directional tuning (compare Fig. 2). For instance, compared with
neurons in the motor cortex in a reaching task, the preferred directions in the cerebellum are more variable in repeated trials (Fortier et al., 1989), neurons in the parietal cortex are less sensitive to static load (Kalaska et al., 1990), and neurons in the
premotor cortex are activated earlier, more transiently (Caminiti et
al., 1991; Crammond and Kalaska, 1996), and affected more frequently by
visual cues (Wise et al., 1992; Shen and Alexander, 1997b). In the
motor cortex and elsewhere, there also exist neurons with complex
properties that are either not task-related or hard to describe but
still could have useful functions in a distributed network (Fetz, 1992;
Zipser, 1992; Moody et al., 1998).
In most of these cases, there are additional free variables besides
hand position. The linear theory may still yield useful results in
these more complex cases after including these additional variables.
For example, the planned movement direction is an independent variable,
which could be used to describe some preparatory activity before overt
hand movement. These new variables should be included when deriving the
tuning rule, as demonstrated in the preceding section by adding the
elbow position as a free variable in abducted reaching.
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REPRESENTING RIGID OBJECT MOTION |
The same geometric argument for arm movement can be applied to
moving rigid objects, which have additional rotational degrees of
freedom around an axis in space (Fig. 1). In the following, we derive a
general tuning rule for rigid motion, discuss its basic properties, and
then contrast the results with concrete models of visual receptive fields.
Description of rigid object motion
Arbitrary instantaneous motion of a rigid object can always be
described by a rotation plus a translation (Fig.
4), but given the same physical motion,
this description is ambiguous up to an arbitrary parallel shift of the
rotation axis. For example, translational velocity can always be
aligned instantaneously with the angular velocity to obtain a screw
motion by passing the rotation axis through the point of zero velocity
in a perpendicular plane (Fig. 4).
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Figure 4.
Arbitrary motion of a rigid object can always be
decomposed instantaneously into a translation and a rotation, allowing
arbitrary parallel shift of the rotation axis. The two examples shown
here describe identical physical motion. Parallel shift of rotation
axis affects the translation velocity but not the angular velocity
.
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This ambiguity disappears when the rotation axis is always required to
pass through the same reference center in the object, say, the center
of mass. We assume that the reference center has been chosen so that a
rigid motion can be described uniquely by a translational velocity and
an angular velocity. We return to this topic later.
The static position and orientation of a rigid object can be specified
by six independent parameters:
|
(24)
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where x, y, z describe the position of the reference
center of the object with respect to a coordinate system fixed to the world, and  1, 2,
3 are three angular variables that represent the
object's orientation. The translational velocity of the object is:
|
(25)
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The angular velocity = (x,
y, z)T in
world coordinates is always linearly related to the time derivatives of
the orientation variables  = (1,
2,3)T:
|
(26)
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where M is an invertible 3 × 3 matrix that
depends only on the orientation (1,
2, 3). For example, when
Euler angles are used to describe orientation (Fig.
5), we have:
|
(27)
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and
|
(28)
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which is invertible as long as det M = sin  0 (Goldstein, 1980).
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Figure 5.
Euler angles (,  ,  ) describe an arbitrary
orientation of a rigid object with axes (X', Y', Z') with
respect to a standard orientation with axes (X, Y, Z).
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Only the abstract linear relation in Equation 26 is needed in the next
section. The actual choice of (1,
2, 3) is unimportant here.
Because the time derivatives of different sets of variables are
linearly related by a Jacobian matrix, Equation 26 always holds regardless of the exact choice of the parameterization of orientation (see also Appendix A on independence of the coordinate system).
Tuning rule for rigid motion
Consider neuronal activity associated with motion of a rigid
three-dimensional object. Assume that the mean firing rate of a
neuron relative to baseline, with a possible time delay, is proportional to the time derivative of a smooth function of the position and orientation of the object in three-dimensional space. In other words:
|
(29)
|
where f is the firing rate, f0
is the baseline rate, and is an arbitrary function of object
position (x, y, z) and orientation (1,
2, 3), as described in the
preceding section. This equation is analogous to Equation 1.
The exact form of function need not be specified here. It may
depend on both the receptive field properties of the cell and the
visual appearance of the object and its surroundings. This formulation
is quite general. For example, all the visual cues of the object
illustrated in Figure 1 are functions of the position and orientation
of the object that completely determine how light is reflected from
various surfaces, whether diffuse (uniform scattering in all
directions) or specular (energy concentrated around the mirror
reflection direction), giving rise to various visual effects such
as shading, shadows, specular reflections, and highlights (Watt and
Watt, 1992). Given that all sensory cues are determined completely by
the position and orientation of the object, we expect a
motion-sensitive neuron to respond to changes of these variables.
The simplest way to estimate these changes is to compute the first
temporal derivative.
The assumption in Equation 29 allows us to derive a general tuning rule
for neurons sensitive to three-dimensional object motion. Given a
three-dimensional object moving at instantaneous translational velocity
v and angular velocity , the mean firing rate
of a generic neuron should depend on these variables in a highly
stereotyped way:
|
(30)
|
where f0 is the background firing rate,
p is the preferred translational direction, given
by:
|
(31)
|
and vector q is the preferred rotation axis,
given by:
|
(32)
|
with matrix M as in Equation 26, and:
|
(33)
|
is an intermediate vector variable, the transformed preferred
rotation axis in the orientation angle space. Both the preferred translational direction p and the preferred rotation axis q are vectors in the physical space. They may depend on the
object and its position and orientation but not on the translational velocity v and angular velocity . The
derivation of Equation 30 follows from the chain rule:
|
(34)
|
where Equations 25 and 26 and the definitions in Equations 31-33
have been used. The derivation of the tuning rule does not depend on
which coordinate system is used (Appendix A).
Before explaining the meaning of the tuning rule in the next section,
first consider the baseline firing rate, which is not constrained by
the present theory and thus requires separate consideration. The
baseline firing rate may itself be modulated by several factors, and
the simplest linear model is:
|
(35)
|
where ai, bi, a,
b are constants, and the position (x, y, z) and the
orientation (1, 2,
3) of the object are included as possibly
relevant factors related to the static view, together with the
translational speed v and the angular speed for object motion, which may also be relevant. This linear equation generalizes Equation 9 for motor neurons. Similarly, Equation 10 can also be generalized by including angular position and speed. This assumes that
the baseline firing rate in general may contain information about both
the static configuration of an object and its instantaneous motion.
Cosine tuning and multiplicative speed modulation
The basic tuning rule in Equation 30 can be rewritten in its
equivalent form:
|
(36)
|
where v = |v| is the speed of
translation, = || is the angular speed of
rotation, p = |p| is the length of the
preferred direction vector, q = |q| is the length of the preferred rotation vector, is the angle between vectors p and v, and  is the angle between
vectors q and .
In other words, given the particular view of a particular object, the
response above baseline should be the sum of two components, one
translational and one rotational. The translational component is
proportional to the cosine of the angle between a fixed preferred translational direction and the actual translational direction. In
addition, it is also modulated linearly by the speed of translation, which does not alter the shape of the tuning curve. Similarly, the
rotational component is proportional to the cosine of the angle between
a fixed preferred rotation axis and the actual rotation axis. In
addition, the rotational component is also modulated linearly by the
angular speed of rotation.
Distribution of preferred direction and preferred axis
Thus far, the view of the given object is assumed to be fixed.
That is, the cosine tunings for both translation and rotation are
defined with respect to a particular view of the object. When the view
of the object changes, the preferred translational direction p and preferred rotation axis q of a
motion-sensitive neuron may also change.
The theory constrains this change because the preferred translational
direction p and the transformed preferred rotation axis
q* are derived as gradient fields in Equations 31 and 33.
Here the intermediate vector q* is related to the preferred rotation axis q in physical space by:
|
(37)
|
according to Equation 32. In three-dimensional space, where curl
is defined, the gradient field implies that any three variables taken
from the six variables (x, y, z, 1,
2, 3) must be
curl-free. For example, when the position (x, y, z) of the
object is fixed, the distribution of the transformed preferred rotation
axis in the orientation space (1,
2, 3) must be
curl-free:
|
(38)
|
Any hypothetical neurons with non-zero curl can be ruled out by
this condition (see below). For a gradient field, the zero curl is
simply attributable to the equality of mixed second partial derivatives
of the potential function, which holds also in higher dimensions. The
equivalent path integral formulation is valid also in all
dimensions:
|
(39)
|
along any closed curve in the six-dimensional space, where
dl = (dx, dy, dz), and d = (d1, d2,
d3). Another equivalent
formulation is that the potential function can be constructed by the path integral:
|
(40)
|
which depends only on the end points, not on the exact path. Here
 = (x, y, z, 1, 2,
3) is an arbitrary point in the parameter
space, and 0 is the value at a given initial point. Therefore, in the gradient theory, how the preferred translational direction and the preferred rotation axis of a neuron change with the
view of a given object cannot be arbitrary but is highly constrained. This can provide testable predictions (see below).
Linear nongradient theory
A more general theory can be obtained by directly assuming a
linear relationship between the firing rate and the components of the
translational velocity v = (, , )
and the time derivatives of the angular variables = (1,
2, 3). This yields the same
tuning rule:
|
(41)
|
where p = (px,
py, pz) and q* = (q*1, q*2,
q*3) are arbitrary vector fields, not
necessarily gradient fields, and Equations 26 and 32 are used in the
last step. This tuning rule gives the same response properties
predicted by the gradient theory for a single view of the object. The
difference shows up when the view changes. The nongradient theory
imposes no constraint on how preferred translational direction and
preferred rotation axis should vary with the view of the object. The
gradient theory is more restrictive, and therefore makes stronger predictions.
Change of reference center
Because the description of the same physical motion of a rigid
object is ambiguous up to a parallel shift of the rotation axis (Fig.
4), we have assumed in the above that the rotation axis always passes
through the same reference point c = (x, y, z) in the
object to ensure uniqueness of description. When a different reference
center c' is chosen, the form of the basic tuning rule in
Equation 30 remains valid, but the preferred rotation axis is affected
in a predictable way:
|
(42)
|
where v' and ' are the translational
velocity and angular velocity for the new reference center
c', and:
|
(43)
|
|
(44)
|
are the new preferred translational direction and rotation axis.
One can readily verify that Equation 42 is valid under Equations 43 and
44, using the relations:
|
(45)
|
|
(46)
|
Therefore, changing the reference center of an object has no
effect on the preferred translational direction of a neuron (Eq. 43),
whereas the preferred rotation axis is altered systematically in a
completely predictable manner (Eq. 44). These relations arise purely
from the ambiguity of the description of rigid motion, and thus apply
to both the gradient and the nongradient theories.
Summary
Simple assumptions have led to a general tuning rule for how the
mean firing rate of a neuron should depend on the instantaneous motion
of an arbitrary rigid object. For each given view of the object, the
firing rate is predicted to be the sum of two terms, one for the
translational motion component and one for the rotational motion
component, both with cosine directional tuning and linear speed or
angular speed modulation. In general, the preferred translational direction and the preferred rotation axis may depend on the identity of
the object as well as its view. This tuning rule is a linear approximation to the geometry of rigid motion and therefore should obtain regardless of the exact computational mechanisms involved. In
other words, this rule is expected to be a robust property for
motion-sensitive neurons responding to realistic moving objects. When
the view of the object changes, both the preferred translational direction and the preferred rotation axis of a neuron may change as
well. The gradient theory provides additional constraints on such
changes, whereas the nongradient theory imposes no further constraints.
As a consequence, these two theories can be distinguished by further
experiments. Finally, although the description of rigid motion is
ambiguous up to a parallel shift of the rotation axis, the effects on
the tuning rule are completely predictable and therefore convey no
additional information about the response properties of a neuron.
Examples of motion-sensitive receptive field models
Many neurons in visual cortex, particularly in the dorsal stream
leading to parietal cortex, respond selectively to visual motion. Here
we consider three-dimensional rigid motion and examine several simple
computational mechanisms that yield explicit analytical formulas for
the preferred translational direction and the preferred rotation axis.
For each fixed view of the object, the results are consistent with the
basic tuning rule in Equation 30. For different views, however, the
global gradient-field condition for the preferred axes can be violated
by the idealized velocity component detectors. This shows that the
neuronal behavior predicted by the gradient theory is not always
identical to that of an optic-flow detector.
Velocity component detectors
As illustrated in Figure
6A, suppose the
firing rate of an idealized neuron detects local motion on the image
plane according to:
|
(47)
|
where f0 is the baseline firing rate,
p is the preferred direction of visual motion, and
u is the local velocity on the image plane inside a small
receptive field of the detector. This local velocity detector resembles
some neurons in the middle temporal area (MT) of monkey, as discussed
in the section after Equation 72.
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Figure 6.
Examples of preferred rotation axis for velocity
component detectors. Visual motion of a finely textured rotating
object, here a sphere, is projected orthogonally onto the image plane.
A, For a single local velocity detector with preferred
direction p, the largest response is elicited by rotation
around the preferred rotation axis q, which is on the
horizontal plane and perpendicular to the vector r c, with c representing the center of the sphere.
B, When the response is the sum of two local velocity
detectors with preferred directions p1 and
p2, the preferred rotation axis
q is perpendicular to the image plane.
|
|
This idealized neuron obeys the basic tuning rule in Equation 30,
namely:
|
(48)
|
in response to a textured rigid object moving at translational
velocity v and angular velocity . Here the preferred direction p is the same constant vector as
in Equation 47, and the preferred rotation axis is given explicitly by:
|
(49)
|
where c is a fixed reference center in the object
(center of the sphere in Fig. 6B), and r is the
coordinate of the point in the object that happens to fall into the
vanishingly small receptive field of the detector. As shown in Figure
6A, the geometry of the situation is quite simple, with the
two orthogonal vectors q and r c both lying on the horizontal plane. For fixed angular
speed, using q as the rotation axis maximizes the response
of the velocity detector.
To derive these formulas, note that a point in the object with
coordinate r has the velocity:
|
(50)
|
on the image plane (Fig. 6A). The desired
equations can be obtained by inserting this into Equation 47 and using
the vector identity:
![<UP><B>p</B></UP> · [<UP><B>ω</B></UP>×(<UP><B>r</B></UP>−<UP><B>c</B></UP>)]=[(<UP><B>r</B></UP>−<UP><B>c</B></UP>)×<UP><B>p</B></UP>] · <UP><B>ω</B></UP>.](/content/vol19/issue8/fulltext/3122/img057.gif)
|
(51)
|
Next, consider a higher-order neuron whose response is the sum of
the outputs of several local velocity component detectors:
|
(52)
|
where pi is the preferred direction
of detector i, and ui is the
image velocity in its receptive field. This neuron also obeys the same
basic tuning rule in Equation 48, with the preferred translational
direction and preferred rotation axis given by:
|
(53)
|
|
(54)
|
For example, when two detectors are arranged as shown in Figure
6B, the preferred rotation axis q is
perpendicular to the image plane, whereas the preferred translational
axis vanishes (p = 0) because the image size
is constant under orthographic projection.
Finally, the basic tuning rule in Equation 48 still holds for image
motion of a rigid object under a perspective projection, which projects
each point (x, y, z) in the real world toward the observer
at the origin (0, 0, 0), leaving an image at (X, Y) in the
image plane at z =  :
|
(55)
|
(Longuet-Higgins and Prazdny, 1980; Heeger and Jepson, 1990). On
the image plane, suppose the velocity component detector i
is located at (Xi, Yi)
with preferred direction pi = (pi1, pi2), then the
preferred direction and preferred rotation axis become:
|
(56)
|
|
(57)
|
where
( i,
 i,
 i) = (xi x, yi y, zi z)/zi with (x, y, z) the reference center of the object, and zi is the
z-coordinate in real world for the physical point that
happens to activate detector i. Unlike the orthographic
projection in Figure 6, this mechanism allows a neuron to respond to
looming or shrinking images, because it does not confine the preferred
translational direction p to the image plane.
Spatiotemporal receptive field
Now consider motion-sensitive linear spatiotemporal receptive
fields that obey the basic tuning rule in Equation 30 with a known
potential function. Let I(X, Y, t) describe the intensity of
an image at location (X, Y) on the image plane at
time t, ignoring color and stereo. Suppose the firing rate
of a neuron with linear receptive field F(X, Y) is
linearly related to how fast the overlap between the image and
receptive field is changing:
|
(58)
|
where the inner product is defined by:
|
(59)
|
This inner product can serve as the potential function postulated
in Equation 29:
|
(60)
|
Therefore, the basic tuning rule in Equation 30 must hold true,
and the preferred translational direction and rotation axis are:
|
(61)
|
|
(62)
|
following Equations 31-33. Here the partial derivatives are with
respect to the implicit variables (x, y, z,
1, 2,
3) for the position and orientation of a
moving object that generates the image.
More generally, consider a neuron with an arbitrary linear
spatiotemporal receptive field G so that its firing rate
is:
|
(63)
|
This becomes identical to Equation 58 for the kernel function:
|
(64)
|
where  is Dirac delta function, which can be approximated by
any narrow and normalized smooth function peaked at the origin. If
Equation 64 is a reasonable approximation to the spatiotemporal receptive field, then the basic tuning rule in Equation 30 as well as
p and q given by Equations 61 and 62 becomes
valid. For nonlinear mechanisms, such as squaring (Adelson and Bergen, 1985) and normalization (Heeger, 1993), the above consideration may
apply only after local linearization.
Existence of a global potential function
In all the concrete examples considered above, the basic tuning
rule in Equation 30 holds true for each given view of an object. However, for a single view, the gradient and nongradient theories are
indistinguishable. By assumption, the nongradient theory allows arbitrary preferred translational direction p and preferred rotation axis q. For a given view, the gradient theory can
also generate any desired constant vectors p and
q from the gradients of the following potential
function:
|
(65)
|
where q* = qM is taken as a constant vector,
c = (x, y, z) is the reference center of the object,
and = (1, 2,
3) describes the orientation of the object.
The gradient theory is globally correct only when a potential function
exists for all views of the object. This is the case for the linear
spatiotemporal model, where the potential function can be given
explicitly (Equation 60). By contrast, for the idealized velocity
component detector, a global potential function in general does not
exist, as shown in Example 1 below.
Because the existence of a potential function does not depend on the
choice of the coordinate system (see Appendix A), we only need to show
that a potential function does not exist in the Euler angle space:
(1, 2,
3) = (, , ), assuming that the center of
the object is fixed. In this three-dimensional space, a potential
function exists if and only if the distribution of the transformed
preferred rotation axis q* is curl free. Now consider two
special examples that do not admit a global potential function:
Example 1: Constant preferred rotation axis fixed to the
world. An explicit example is the model in Figure 6B,
where the preferred rotation rotation axis q is the same
regardless of the orientation of the spherical object. Here it is
assumed that the velocity component detector has two vanishingly small
receptive fields that can nevertheless detect the true local velocity
components regardless of the orientation of the object. Without loss of
generality, take the preferred rotation axis as a unit vector in the
negative Y-axis:
|
(66)
|
and then compute its counterpart vector in the Euler angle space:
|
(67)
|
It is verified that curl q* as defined in Equation 38
does not vanish. This proves that the desired potential function cannot
exist for this hypothetical neuron (Fig.
7).
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Figure 7.
The gradient theory prohibits a neuron from having
a constant preferred rotation axis in the physical space regardless of
the view of the object, because this would allow the transformed
preferred rotation axis (q*1,
q*2, q*3) to
have non-zero curl in the Euler angle space (, , ). Here the
constant preferred rotation axis q = (0, 1, 0) is
taken as the negative Y-axis. Only two-dimensional
visualization is needed here because q*2 0, and the vector field is independent of .
|
|
Example 2: Constant preferred rotation axis fixed to the
object. A possible example is a vestibular neuron receiving input from only a single semicircular canal without other influences such as
that from the otolith. Then the firing rate has a cosine tuning with
respect to a preferred rotation axis fixed on the head of the animal,
regardless of the orientation of the head in the world (Baker et al.,
1984; Graf et al., 1993). Without loss of generality, let the preferred
rotation axis be a unit vector in the positive Z' axis of
the object (head), then in world coordinates this axis is:
|
(68)
|
so that its counterpart vector in the Euler angle space is:
|
(69)
|
Here curl q* is not zero, proving the nonexistence of
the desired potential function.
Therefore, there are simple computational mechanisms that can violate
the gradient theory. As shown above, the gradient theory prohibits a
neuron from having a truly invariant preferred rotation axis fixed
either to the world or to the object. These neurons, however, are
allowed by the nongradient theory. In particular, the prediction of the
gradient theory can differ from that of an optic-flow sensor. As
another example, the preferred translational direction field for a
small moving dot may also have non-zero curl when measured at different
regions inside a large MST-like receptive field that has circular
arrangement of local preferred directions (Saito et al., 1986). These
idealized examples demonstrate that the global property of the gradient
theory is quite restrictive, which, however, makes its prediction
strong and refutable. Experiments could be performed to test whether
the gradient theory accounts for the neuronal responses to
three-dimensional object motion.
| |
COMPARISON WITH EXPERIMENTAL RESULTS OF SINGLE NEURONS |
The tuning rule for reaching arm movement in Equation 2 is a
special case of the general tuning rule in Equation 30 without the
rotational terms. Biological evidence from the motor system in support
of the tuning rule has already been considered in the preceding
sections. In this section we examine several additional biological
examples that are consistent with some special cases of the general
tuning rule and then discuss more comprehensive tests for moving rigid objects.
One-dimensional example: hippocampal place fields on a
linear track
For one-dimensional movement, the linear tuning theory predicts
only linear speed modulation, without further constraint on the tuning
function. The firing rate is given by:
|
(70)
|
where x is the variable of interest, v = dx/dt is the speed, and (x) is the gradient of
function . This is a special case of the general tuning rule in
Equation 36, keeping only the translational term, with cos = 1 and
p = (x). Function (x) is allowed to be
completely arbitrary because a potential function:
|
(71)
|
can always be constructed. For example, the firing rates of
hippocampal place cells are modulated by running speed when a rat moves
on a narrow track (McNaughton et al., 1983). In this one-dimensional
problem, Equation 70 does not constrain the tuning function
(x), here interpreted as a place field, describing the mean firing rate at spatial position x. When averaged over a
population of simultaneously recorded place cells to get enough spikes,
the average firing rate was indeed remarkably linearly related to the
running speed, with a correlation coefficient >0.95 over the full
range of speeds (Zhang et al., 1998), in agreement with Equation 70. A
similar linear relationship was also reported recently for rats on a
running wheel (Hirase et al., 1998). The theory gives a correct tuning
rule without reference to the underlying biological mechanisms.
Two-dimensional example: local translational visual motion
Neurons in middle temporal area (MT or V5) of monkeys respond
selectively to the direction of local visual motion (Zeki, 1974; Maunsell and Van Essen, 1983; Albright, 1984), although they are also
affected by other factors, such as surround motion (Allman et al.,
1985; Tanaka et al., 1986; Raiguel et al., 1995), pattern motion
(Movshon et al., 1985), transparency (Stoner and Albright, 1992; Qian
and Andersen, 1994), and form cues (Albright, 1992). Consider the
following formula obtained by keeping only the translational term in
Equation 30:
|
(72)
|
which has been used for reaching arm movement (Eq. 2). This tuning
rule also resembles the directional sensitivity of MT neurons (Zhang et
al., 1993; Bura as and Albright, 1996), where p is the
preferred direction of the neuron, and v is the velocity of
the stimulus inside the receptive field.
This simple formula can capture two primary features of many MT
neurons: a broad directional tuning curve, and speed modulation without
changing the shape of the tuning curves (Rodman and Albright, 1987),
while setting aside various other properties accounted for by more
detailed models (Sereno, 1993; Nowlan and Sejnowski, 1995;
Buraas and Albright, 1996; Simoncelli and Heeger, 1998). For
many MT neurons, the tuning curves are often sharper than cosine, in
which case a circular normal curve in Equation A17 might provide better
fit because of its closeness to a Gaussian (Albright, 1984). Linear
speed modulation is probably a reasonable approximation for some
neurons when the velocity is slow, but typically firing rates often
decrease after reaching a peak at an optimal speed (Maunsell and Van
Essen, 1983). It would be interesting to test whether speed modulation
is linear when averaged over raw firing rates for a large population of
neurons, especially under ecologically plausible stimulus conditions.
The above consideration may also apply to many V4 neurons, which
responded to visual motion response in an MT-like manner (Cheng et al.,
1994). Cosine tuning curves for translational motion have also been
described in the cerebellum (Krauzlis and Lisberger, 1996) and the
parietal area 7a (Siegel and Read, 1997).
Three-dimensional object motion
Spiral motion
No direct experimental data are available on how a neuron responds
systematically to a realistic moving three-dimensional object with
arbitrary translation and rotation. One closely related example is the
broad tuning of some neurons to spiral visual motion, which may be
generated plausibly by a large moving planar object facing the observer.
As shown in Figure 8, neurons in monkey
visual medial superior temporal area (MST), which receive a major input
from area MT, typically respond well to wide-field random-dot spiral
motion patterns (Graziano et al., 1994). Most neurons in the ventral intraparietal area (VIP) are also sensitive to visual motion (Colby et
al., 1993), and some have tuning properties to spiral motion similar to
those in area MST (Schaafsma and Duysens, 1996), probably due to input
directly from MST and/or integration of inputs from area MT. Area 7a is
at a higher level than MST and might have more complex response
properties for optic flows (Siegel and Read, 1997). In theory, it is
possible to build an MST-like neuron from MT-like local motion inputs,
even with position-invariance properties (Saito et al., 1986; Poggio et
al., 1991; Sereno and Sereno, 1991; Zhang et al., 1993). Tuning to
spiral motion was predicted based on Hebbian learning of optic flow
patterns (Zhang et al., 1993) and by other unsupervised learning
algorithms (Wang, 1995; Zemel and Sejnowski, 1998).
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Figure 8.
Visual cortical neurons sensitive to spiral optic
flow may provide evidence for the existence of preferred translation
direction and preferred rotation axis for three-dimensional object
motion. Here the average response of a neuron depended on the
combination of radial and circular motion components, which
varied systematically by changing local velocity directions in the
moving random dot stimulus. The solid curves are circular
normal fit, and the dashed curves are cosine fit. Data of
the MST neuron are from Figures 8C in Graziano et al. (1994), and the
VIP neuron data are from Figure 10 in Schaafsma and Duysens (1996),
with permission.
|
|
To explain spiral tuning in terms of rigid motion, regard the
environment itself as a large rigid object, moving relative to the
observer. For the experiments mentioned above, the environment may be
considered as a finely textured screen, oriented vertically, facing the
observer. Translating this screen toward or away from the observer
induces expansion or contraction, whereas rotating the screen around a
perpendicular axis induces circular motion. According to the basic
tuning rule in Equation 36, a neuron should respond to arbitrary motion
of this screen with the firing rate:
|
(73)
|
where v is the translational speed, is the angular
speed, and:
|
(74)
|
are constants, where is the angle between the
preferred translational direction and the actual translational
direction, and is the angle between the preferred rotation axis and
the actual rotation axis.
To see why this accounts for spiral tuning, write Equation 73 in the
equivalent form:
|
(75)
|
where s = (v, r) is the spiral
composition vector describing the stimulus, w = (P,
Q/r) is the preferred spiral composition vector
describing response properties of the neuron, r is a
constant length, introduced to make v and r of
the same units, and  is the angle between s and
w. In the polar diagrams in Figure 8, the horizontal and
vertical axes correspond to the two components of the stimulus spiral
composition s, whose length:
|
(76)
|
was fixed during the experiments. It follows from Equation 75 that
the response should fall off smoothly when the stimulus spiral
composition becomes different from the preferred composition, in
proportion to the cosine of the angle between them. This broad
tuning to spiral motion is generally consistent with the data (Graziano
et al., 1994; Schaafsma and Duysens, 1996), although a circular normal
function in Equation A17, with one more free parameter, may provide
better fits than a cosine function (Fig. 8).
The above interpretation implies that firing rate should scale linearly
with translational speed and angular speed independently of the spiral
tuning curve. The responses of most MST neurons do indeed depend on
speed (Tanaka and Saito, 1989; Orban et al., 1995), and many are
monotonically increasing (Duffy and Wurtz, 1997a). It would be
interesting to test quantitatively how well the linearity holds when
averaged over a population of cells, especially for an ecologically
relevant range of motion.
The tuning rule in Equation 73 also implies that the response should
depend on the focus of expansion or the translational component in the
optic flow, which also occurs for many MST cells (Duffy and Wurtz,
1995, 1997b). Adding a translational velocity vector to the stimulus
corresponds to translating the stimulus screen sideways, which affects
the angle in Equation 74 and thus the response in Equation 73.
Changing the translational direction and the rotation axis of the
stimulus screen can alter both angles and in Equation 74 and
thus the predicted response in Equation 73.
Motion-sensitive neurons in area MST may be used for purposes such as
estimating heading or self-motion (Perrone and Stone, 1994; Lappe et
al., 1996) or segmenting multiple moving objects (Zemel and Sejnowski,
1998). MST responses can be affected by various factors, including, for
example, surround motion (Tanaka et al., 1986; Eifuku and Wurtz, 1998),
disparity (Roy et al., 1992), eye position and movement (Newsome et
al., 1988; Bradley et al., 1996; Squatrito and Maioli, 1997),
vestibular input (Thier and Erickson, 1992), form cues (Geesaman and
Andersen, 1996), the presence of multiple objects (Recanzone et al.,
1997), and attention (Treue and Maunsell, 1996). Most experiments used
simplified stimuli, although more realistic stimuli were tested
recently (Sakata et al., 1994; Pekel et al., 1996). Because most of
these examples contain parameters other than the object's position and orientation, additional variables are needed to account for all of
these effects in a model.
Further experimental test
Given all the contributing factors mentioned above, it is natural
to ask how a neuron would respond to a more natural-moving three-dimensional object. A simple geometric stimulus is easier to
specify and present but may lack important sensory cues needed to
predict the response of a neuron to a natural stimulus. Our analysis
relies on varying the translational direction and rotation axis and
might provide a convenient basic description for response properties in
terms of a preferred translational direction and a preferred rotation axis.
To test directly the basic tuning rule in Equation 30 or 36, one should
present realistic images of a moving three-dimensional object to
motion-sensitive neurons. The simplest way to test the theory is to
oscillate slightly an object around a fixed axis. The oscillation
should be sufficiently small so that salient visual cues are not
occluded. For sinusoidal oscillations with frequency  and amplitude
 :
|
(77)
|
and the angular speed is = d/dt in Equation 36.
Thus the theory predicts that the firing rate as a function of time
should be:
|
(78)
|
where is the angle between the actual rotation axis and the
preferred rotation axis, q is a constant coefficient, and
 is the latency for visual response. The translational term in Equation 36 vanishes because there is no translational motion
(v = 0). Systematically changing the orientation of the
rotation axis (angle ) while keeping the view fixed allows the
tuning function and the preferred rotation axis to be measured for the neuron.
Similarly, the response to translation in three-dimensional space could
be tested by oscillating the whole object along a straight line:
|
(79)
|
The translational speed is v = dx/dt in Equation 36, and the firing rate is:
|
(80)
|
where is the angle between the actual translational direction
and the preferred translational direction, and p is a
constant coefficient. The preferred translational direction can be
measured by systematically changing the translational direction (angle ) while keeping the view fixed.
For more efficient tests, the object could be rotated continuously with
varying angular speed, covering all relevant views, first with respect
to a fixed axis, and then systematically changing the axis. If the
basic tuning rule is correct and the system is essentially linear, the
tuning function and the preferred rotation axis could be computed for
each view of the object. An even more efficient test is possible with a
continuously time-varying rotation axis that generates tumbling
movements of the object (Stone, 1998).
Eye position is one implicit factor that may affect the preferred
translational direction and preferred rotation axis. The present theory
allows an eye position effect but provides no additional constraints.
The linear response properties of a neuron for a given object are
specified completely by its preferred translational direction, preferred rotation axis, and baseline firing rate for each given view
of the object, as well as how these parameters depend on the view. All
of these properties are experimentally testable and can be compared
with the theoretical predictions in the preceding sections. For
example, with the center of the object fixed, the curl-free
condition for a given neuron should be tested by measuring its
preferred rotation axis for four or more different orientations of the
object. For a full test in six-dimensional space, both the preferred
translational direction and the preferred rotation axis of the neuron
should be measured for seven or more different positions and
orientations of the object. See Appendix B for further discussion.
If motion-sensitive neurons with similar tuning properties are
clustered in the brain, then it might be possible to use functional magnetic resonance imaging techniques to test the predicted properties of the tuning rule in animal and human subjects using realistic images
of moving 3-D objects as visual stimuli.
| |
DISCUSSION |
An explanation for cosine tuning
The remarkable ubiquity of approximately cosine tuning curves for
a wide range of neural responses in the visual and motor systems
suggests that there may be a common explanation that transcends the
specific mechanisms that generate these response properties. We have
shown that the low dimensionality of the geometric variables that
underlie object motion and body movements could account for these
observations. The gradient formulation of this general principle provides a rigorous framework for unifying the dependence of tuning curves on the axes and speeds of rotation and translation.
This theoretical framework makes a number of specific predictions. The
primary prediction is the existence of preferred axes of rotation and
translation for moving objects, which can be determined by
systematically rotating and translating objects in the receptive field
of cortical neurons. The firing rate of a neuron should fall off in
proportion to the cosine of the angle between the preferred rotation
axis and translational direction and the true rotation axis and
translational direction. In addition, the speed and angular speed
should modulate the firing rate multiplicatively without changing the
shape of the directional tuning functions, somewhat related to the
multiplicative gain fields in the parietal cortex for eye position
(Andersen et al., 1997; Salinas and Abbott, 1997) and recent evidence
for distance modulation of responses in visual cortex (Trotter et al.,
1992; Dobbins et al., 1998). A secondary prediction is that the fields
of preferred directions of rotation and translation for each individual
neuron are curl free; these are global conditions on the overall
pattern of vectors. The curl-free condition is relaxed in nongradient
theory, which provides a theoretical alternative that can be
experimentally tested.
Cosine tuning for the direction of arm movement characterizes many
neurons in the motor cortex as well as in other parts of the motor
system. If this cosine tuning with direction mainly reflects the
geometrical constraint of moving in a three-dimensional space, as we
propose, then the specific functions of these neurons in guiding and
planning limb movements must be sought in other properties. One way to
obtain this information is to measure how the preferred direction
varies in space for different hand positions, because this vector field
completely specifies the properties of a neuron in a linear theory.
When the preferred direction field is curl free, the underlying
potential function that generates the gradient field can be constructed empirically.
Cosine tuning function is an approximation to biological data. For
example, the averaged directional tuning curves in Figure 2 are all
slightly sharper than a cosine function. Such systematic deviation can
be accounted for only by nonlinear theories (see Appendix A).
Ultimately, the underlying neural mechanisms that generate the tuning
properties need to be considered in more detailed theories. These
tuning properties might be the outcome of learning processes based on
correlated neuronal activities induced by movement and motion. In this
paper, we have focused on several analytically tractable situations to
emphasize the existence of general neuronal tuning properties that are
insensitive to the actual mechanisms.
How preferred rotation axes may be used to update a
static representation
If the cosine tuning of motion-sensitive neurons is determined
essentially by geometric constraints regardless of the actual computational functions, then what is the value of these simple response properties? For three-dimensional object motion, a population of neurons tuned to translational direction and rotation axis should
carry sufficient information to determine the instantaneous motion of
any given object and therefore could be used to update the static view
represented elsewhere. This allows future sensory and motor states to
be predicted from the current static state.
Information about the static view of an object is
represented in the ventral visual stream in the monkey cortex, leading
to the inferotemporal (IT) area (Ungerleider and Mishkin, 1982). The
response of a view-sensitive neuron in IT area typically drops off
smoothly as the object is rotated away from its preferred view, around
either the vertical axis (Perrett et al., 1991) or other axes
(Logothetis and Pauls, 1995). View-dependent representations for
three-dimensional objects have been studied theoretically (Poggio and
Edelman, 1990; Ullman and Basri, 1991) and have motivated several
recent psychophysical experiments (Edelman and Bülthoff, 1992;
Bülthoff et al., 1995; Liu et al., 1995; Sinha and Poggio, 1996).
The general idea of a view-dependent representation in the IT area is
consistent with recent neurophysiological results, including
single-unit recordings (Perrett et al., 1991; Logothetis and Pauls,
1995; Logothetis et al., 1995) and optical imaging data (Wang et al.,
1996).
Information about the instantaneous motion of an object is
represented in the dorsal visual stream, including areas MT, MST, superior temporal polysensory area, and the parietal cortex, such as
area 7a. Given a population of motion-sensitive neurons tuned to
translation and rotation, it should be possible to extract complete
information about the instantaneous motion of any object. For example,
a six-dimensional population vector can be used to reconstruct rigid
motion. More efficient reconstruction methods may also be used and
implemented by a biologically plausible feedforward network (Zhang et
al., 1998). The same set of neurons can extract the motion of different
objects by combining the activities of input neurons differently.
Instantaneous translation and rotation determine how the current view
of this object is changing at the moment and could be used to update
the static view representation in the IT area.
Broad tuning to static views logically implies that each static view of
an object elicits a certain activity pattern in the temporal cortex,
and that as the view changes, the pattern of activity also changes
smoothly, depending on the axis of rotation and direction of
translation. A complete representation of the dynamic state of an
object would require representing information about both the current
view and how the view is changing, so that the system can
effectively update its internal state in accordance with the
movement of the object. Such motion information might help improve
the speed and reliability of the responses of view-specific neurons to
a three-dimensional object during natural movements.
Conclusion
We have shown that simple generic tuning properties arise when an
encoded sensory or motor variable reflects changes rather than static
configurations. By linearizing the system locally for
movement-sensitive neurons, the analysis reveals mechanism-insensitive tuning properties that mainly reflect the geometry of the problem rather than the exact encoding mechanisms, which could be much more
complicated. Although a nonlinear analysis is also considered (Appendix
A), the basic linear theory already captures some essential features of
the biological data, such as sensory responses to visual pattern
motions and directional tuning for reaching movements. The analysis
predicts the existence of a preferred translational direction and a
preferred rotation axis in space with cosine tuning functions for
representing arbitrary three-dimensional object motion. For natural
movements that have an intrinsically low dimensionality, combinations
of variables become highly constrained and cannot be changed
arbitrarily. It is precisely for these constrained movements that the
mechanism-insensitive properties studied here may become useful. By
contrast, the analysis may not apply for artificial movements such as
computer-generated visual motion stimuli that do not satisfy any
simplifying geometry constraints that occur in the real world. The
brain should have more efficient representations for those stimulus
features that are consistent with commonly encountered configurations
in the real world. The analysis presented here may help to predict
tuning properties of motion-sensitive neurons in unknown situations by
providing a basic description of expected properties with which more
detailed characterizations as well as potential deviations can be contrasted.
| |
FOOTNOTES |
Received Oct. 16, 1998; revised Dec. 14, 1998; accepted Jan. 27, 1999.
We are grateful to T. D. Albright, G. T. Buraas,
G. E. Hinton, R. J. Krauzlis, K. D. Miller, A. B. Schwartz, M. I. Sereno, M. P. Stryker, R. S. Turner, D. Zipser, and two anonymous reviewers for helpful comments on the
analysis presented here.
Correspondence should be addressed to Dr. Terrence Sejnowski,
Computational Neurobiology Lab, The Salk Institute, La Jolla, CA 92037.
| |
APPENDIX A: EXTENDED THEORIES |
We first reformulate the linear tuning theory for motion-sensitive
neurons in general terms and then make nonlinear extensions. Here it is
assumed that the natural movements of interest can be parameterized by
a low-dimensional vector variable:
|
(A1)
|
with D degrees of freedom. For example, D = 3 for reaching with a stereotypical arm posture, and D = 6 for rigid object motion. D can be larger when more
independent variables are included.
Linear gradient theory
Assume that the mean firing rate of a motion-sensitive neuron is
linearly related to the time derivative of a potential function (x) of the state variable x. This leads to the tuning rule:
|
(A2)
|
where f0 is the baseline rate, the
gradient:
|
(A3)
|
is the generalized preferred direction, and:
|
(A4)
|
is the generalized velocity. A gradient is often treated as a
vector, although mathematically it is more properly called a one-form
(Flanders, 1989), a distinction, however, that has no practical
consequence for this paper. The necessary and sufficient condition for
the preferred direction field g = (g1, g2, ... ,
gD) to be a gradient field is that:
|
(A5)
|
This is equivalent to the condition that:
|
(A6)
|
for any closed path, or that the potential function can be
reconstructed from the preferred direction by:
|
(A7)
|
where the integral depends only on the end points x and
x0, not on the path. In three-dimensional
space, these conditions are equivalent to zero curl.
Linear nongradient theory
Assume directly a linear relationship between the firing rate and
the components of the generalized speed velocity v. This
leads to the tuning rule:
|
(A8)
|
without further constraint on the distribution of the preferred
direction g(x). In particular, it may not be the
gradient of a potential function. Thus, in general:
|
(A9)
|
and the path integral may depend on the path. For a fixed
x, the local tuning property of this neuron is the same as
that predicted by the gradient theory. The difference is that the
nongradient theory allows an arbitrary global distribution of the
preferred direction field, whereas the gradient theory admits only a
gradient field.
Coordinate-system independence
Both the gradient and the nongradient theories are
independent of which variables are chosen to parameterize the
movements. Suppose the old vector variable x and a new
variable  are related by:
|
(A10)
|
then the velocities v = dx/dt and
 = d/dt in the two
coordinates are related linearly by:
|
(A11)
|
where J() = dh()/d is the Jacobian matrix.
The tuning rule has the same form in both coordinate systems:
|
(A12)
|
where the new preferred direction is:
|
(A13)
|
If a potential function exists such that the old preferred
direction is a gradient field:
|
(A14)
|
then the preferred direction field is still a gradient field with
respect to the new variable :
|
(A15)
|
where the new potential function is:
|
(A16)
|
The choice of variable is arbitrary, but convenience favors
coordinates that make the results easiest to interpret.
Nonlinear theory: circular normal tuning
The circular normal tuning function for firing rate has the
general form:
|
(A17)
|
which has one more free parameter and tends to fit data better
(compare Figs. 2 and 8) than a cosine tuning function of the general
form:
|
(A18)
|
where is the angular variable of interest, and A, B, K,
A', B' are parameters.
The circular normal function can mimic either a cosine or a Gaussian.
When K is very small, exp(K cos ) ~ 1 + K cos so that the circular normal function in Equation A17 approaches the cosine function in Equation A18 with A' = A + B and B' = BK. When K is large, cos
~ 1 2/2 so that the circular normal
function approaches a narrow Gaussian function with the variance
1/K.
How can we generate a circular normal tuning function? Because the
time-derivative equation for firing rate:
|
(A19)
|
can yield a cosine tuning function, as shown before, the modified
equation of the form:
|
(A20)
|
should yield a circular normal tuning function, where a
and b are parameters. The result is a tuning rule of the
form:
|
(A21)
|
where angle specifies movement direction and v is
movement speed. This modification introduces the following testable
effects on speed modulation. If parameter p is very small,
the tuning function is close to a cosine, and the firing rate increases
approximately linearly with speed, as before. When the parameter
p is larger, the tuning function is closer to a Gaussian,
and the firing rate should increase faster than linearly with speed.
The directional tuning also becomes narrower with higher speed. These
effects are similar to those shown in Figure
9A for quadratic terms. See related discussion in the next section.
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|
Figure 9.
Effects of quadratic terms on directional tuning
curve and speed modulation, here showing the tuning functions
f = 2 + 10(v + v cos ± v2 cos 2), which have a
single peak when the speed v  0.25, with the critical
cases indicated by  . A, Positive sign: As the speed
increases, the tuning curve becomes progressively narrower, and its
peak response grows more slowly than a linear function (dashed
line). B, Negative sign: As the speed increases, the
tuning curve becomes progressively wider, and its peak response grows
more slowly than a linear function (dashed line).
|
|
Although Equation A20 can lead to a circular normal function, it does
not specify how this occurs. One plausible biological mechanism is
a recurrent network with appropriate lateral connections, which can
generate a tuning curve closer to a circular normal function than to a
cosine function (Pouget et al., 1998).
Nonlinear theory: acceleration tuning and quadratic
speed modulation
In this section the basic tuning theory is generalized by
including the second temporal derivative. This leads to acceleration tuning, nonlinear speed modulation, and departure from perfect cosine
directional tuning.
Assume that the firing rate of a neuron contains not only the first
temporal derivative of a potential function, but also the second
temporal derivative of another potential function:
|
(A22)
|
where f0 is the baseline rate, = (x) and  = (x) are two unknown potential
functions of a vector variable x = (x1,
x2, ... , xD). The
special case = 0 reduces to what has been considered before.
This assumption leads to the new tuning rule:
|
(A23)
|
where v = (v1,
v2, ... , vD) = dx/dt is generalized velocity, a = dv/dt is generalized acceleration, p1 = is the preferred direction
for velocity, p2 = is the
preferred direction for acceleration, and hij =  2/xixj is the
Hessian matrix of second derivatives.
Example 1: Reaching movement
Here the vector variable x = (x1,
x2, x3) = (x, y, z)
describes the hand position. The dot product terms in Equation A23 mean
that both the velocity and the acceleration have preferred directions
and cosine directional tuning functions, together with multiplicative
linear modulation by the speed or the magnitude of acceleration. Ashe
and Georgopoulos (1994) included acceleration terms in a different
regression formula and found a small number of cells related to hand
acceleration. Systematic tests are needed to determine whether the
acceleration tuning predicted by Equation A23 really exists.
Example 2: Rigid object motion
Here the vector variable x = (x1,
x2, ... , x6) = (x, y, z,
1, 2,
3) describes the object's position and
orientation in space. By transforming
(1,
2,
3) into the angular velocity in physical space, the tuning rule becomes:
|
(A24)
|
where v = (, , ) is the
translational velocity, a = dv/dt is the
translational acceleration, = (1,
2, 3) is the angular
velocity, = d/dt is the angular
acceleration, and Hij, Iij,
Jij are determined by the Hessian matrix and matrix
M as in Equation 26. Therefore, angular acceleration should
also have a cosine tuning with a preferred direction
q2. Although there is some evidence
that some MT cells can provide information about acceleration of a
visual target (Movshon et al., 1990; Lisberger et al., 1995),
systematic tests for acceleration tuning with realistic objects have
not been performed.
Effects of quadratic terms
The quadratic speed terms imply both nonlinear speed modulation
and higher-order Fourier components for directional tuning that
are speed dependent. To see this, consider a two-dimensional reaching
example with the hand velocity:
|
(A25)
|
where v is the speed and is the angle of movement
direction. The quadratic terms can always be written as:
|
(A26)
|
where h12 = h21, and the
coefficients a, b and the phase shift 0
depend only on hij. For comparison, the linear
speed term can be written as
p1 · v = pv cos( + 1), with a phase shift 1.
The second Fourier component can either sharpen or broaden the original
cosine function, depending on its sign. As illustrated in Figure 9, if
the tuning curve is sharpened by the second component, it becomes even
sharper as the speed increases; if the tuning curve is broadened, then
it becomes even broader as the speed increases. However, the amplitude
of the second component (cos 2) should be no more than one-fourth of
that of the first one (cos ) to ensure that the tuning curve has
only a single peak. This limits the effects from the second Fourier
term. For speed modulation, the quadratic speed factor produces only a
slight bend (Fig. 9) and is too weak by itself to produce  -shaped or  -shaped curves for some neurons in area MT (Maunsell and Van Essen,
1983; Rodman and Albright, 1987) and MST (Orban et al., 1995; Duffy and
Wurtz, 1997a). Rodman and Albright (1987) also showed that the average
tuning widths of MT neurons were insensitive to speed, although the
typical experimental errors for individual neurons might mask the small
effects shown here. Thus, the second Fourier component with squared
speed may help improve data fitting (compare Fig. 2), but probably only
within a narrow range of speeds.
| |
APPENDIX B: LINEAR VECTOR FIELD FOR DATA ANALYSIS |
Linear vector field from experimental data
As shown in the main text, the preferred direction and the
preferred rotation axis may be generated by gradient fields of a
potential function. Here we consider how to test the gradient condition
experimentally. In two and three dimensions, where the curl can be
defined, a vector field generated as the gradient of any potential
function is always curl free (Fig. 10).
In particular, a two-dimensional vector field (u(x, y), v(x,
y)) has both zero curl and zero divergence if and only if
|
(B1)
|
is analytic or differentiable with respect to the complex variable
z = x + iy, because of the Cauchy-Riemann
differentiability conditions. One can use templates of curl-free vector
fields generated by gradient of potential functions to interpolate
arbitrary vector fields (Mussa-Ivaldi, 1992). Therefore, the curl-free
condition alone is too flexible because it can locally distort fields
to accommodate any sparsely sampled vector field data.
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Figure 10.
Curl-free vector fields may have various visual
appearances. These examples, obtained as the gradients of the potential
functions as indicated, have zero curl everywhere, except at a singular
point in the two cases at the bottom. Plot range: always from 1 to 1 for both axes.
|
|
For testing the gradient-field condition or the curl-free condition
with sparsely sampled data points, an additional smoothness constraint
is needed. Linearity is a reasonable smoothness requirement, at least
for a local region, such as in measurement of local force fields
(Mussa-Ivaldi et al., 1985; Giszter et al., 1993), and in local optic
flow analysis (Koenderink and van Doorn, 1976). A linear vector field
has the general form:
|
(B2)
|
where the column vector x = (x1,
x2, ... ,
xD)T contains the variables of
interest, A is a constant matrix:
|
(B3)
|
and b is a constant vector. If the linear vector field
is the gradient of an unknown potential function, then this function
should have the general quadratic form:
|
(B4)
|
which has the gradient:
|
(B5)
|
with b a constant column vector, and T for transpose.
To be consistent with Equation B2, matrix A must be
symmetric:
|
(B6)
|
in which case we can set B = A. This
symmetry condition in Equation B6 is the necessary and sufficient
condition for a linear vector field to be a gradient field. It includes the curl-free condition for D = 3 as the special case,
because the curl:
|
(B7)
|
vanishes if and only if the matrix A is symmetric.
To determine A and b from data vectors
p1, p2,
... , pN sampled at positions
r1, r2,
... , rN, respectively, we require the total number of data points:
|
(B8)
|
in D-dimensional space, because matrix A and
vector c contain D2 + D scalar
unknowns, whereas N data points provide DN scalar values. For example, the two-dimensional case requires a mini-mum of
three data points, and the three-dimensional case requires a minimum of
four data points.
The least-square solution is:
|
(B9)
|
where
|
(B10)
|
is the Moore-Penrose pseudoinverse, and the matrices are defined
as:
![<UP><B>P</B></UP>=[(<UP><B>p</B></UP><SUB>1</SUB>−<UP><B>p</B></UP><SUB>0</SUB>), (<UP><B>p</B></UP><SUB>2</SUB>−<UP><B>p</B></UP><SUB>0</SUB>),…, (<UP><B>p</B></UP><SUB>N</SUB>−<UP><B>p</B></UP><SUB>0</SUB>)],](/content/vol19/issue8/fulltext/3122/img129.gif)
|
(B11)
|
![<UP><B>X</B></UP>=[(<UP><B>x</B></UP><SUB>1</SUB>−<UP><B>x</B></UP><SUB>0</SUB>), (<UP><B>x</B></UP><SUB>2</SUB>−<UP><B>x</B></UP><SUB>0</SUB>),…, (<UP><B>x</B></UP><SUB>N</SUB>−<UP><B>x</B></UP><SUB>0</SUB>)],](/content/vol19/issue8/fulltext/3122/img130.gif)
|
(B12)
|
which are made of column vectors as shown, and:
|
(B13)
|
|
(B14)
|
|
(B15)
|
The above solution minimizes:
|
(B16)
|
which vanishes only when all the data points can be fit exactly by
the linear model.
Examples of the simplex method
In this section, we illustrate an alternative formulation of the
curl-free condition with minimal data points in three-dimensional space, which can be extended readily to other dimensions. For local
interpolation in three-dimensional space, a vector field should at a
minimum be sampled at four locations 1, 2, 3, 4, not all lying in the
same plane (Fig. 11). Suppose the
coordinates of the four points are x1,
x2, x4,
x4, and the corresponding data vectors
are p1, p2,
p3, p4. The
simplex method is based on the fact that any point x in
three-dimensional space can be expressed as:
|
(B17)
|
where the coefficients are unique, satisfying the constraint:
|
(B18)
|
The linearly interpolated vector at this position is:
|
(B19)
|
Similar equations with fewer variables hold for the one- and
two-dimensional cases.
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Figure 11.
Interpolating a vector field linearly in
three-dimensional space requires at least four data vectors to be
measured at locations not all in the same plane. In two-dimensional
space, at least three data vectors are required at locations not all
from the same line. The case shown at the bottom is degenerate.
|
|
To derive an integral formula for the curl-free condition, first
integrate along the straight line segment from
x1 to x2 with a linearly
interpolated vector, yielding:
|
(B20)
|
To evaluate the closed path integral along the triangle 123 in Figure 11, use this formula and the equality
a1 + a2 + a3 = 0, with a1 x3 x2, and so on, to obtain:
|
(B21)
|
which must vanish if the field is curl free. This formula is valid
in both two- and three-dimensional cases. For the tetrahedron in Figure
11, a similar closed path integral along the edges should vanish.
Because there are only three independent loops, we can choose, for
instance:
|
(B22)
|
Although equivalent to the matrix symmetry in Equation B6, the
path integral formulas are more intuitive and express the curl-free condition in terms of directly measurable quantities, as in Equation B21.
In the experiment by Caminiti et al. (1990), the preferred direction of
a motor cortical neuron was sampled at three points at equal distance,
similar to the case in the bottom diagram in Figure 11. Here the
linearity of the vector field only entails that:
|
(B23)
|
which is a special case of Equation B19. The curl-free condition
cannot be tested here. With linear interpolation, at least four
starting hand positions should be sampled (Fig. 11).
| |
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ABSTRACT
INTRODUCTION
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